The article is organized as follows:
In Section  we introduce the initial value problem given by (REF ) following the general framework studied in [1]} and deduce the corresponding dispersion relation.
In Section  the dispersion relation is studied in detail and the asymptotics at low and high frequencies clearly exhibit the scale effects ruled by nonlocality
(see Theorems REF  and REF ). In particular one sees that at low frequencies,
hence at large physical scales, the propagation is quite similar to that governed by the classical wave equation, while at high frequencies, hence at small physical scales, the propagation is remarkably different, due to nonlocality. Furthermore, the analysis of the derivatives of the dispersion reveals new and somehow unexpected features. Indeed (see Theorem REF ), due to nonlocality these derivatives present exotic decay with highly oscillatory behavior at high frequencies. Since these quantities are related with the velocity of energy transport, this suggests that some pieces of information
could be hidden at small scales because of the strange behavior of their propagation velocity.
In Section  we prove some decay properties of the solution of the peridynamic problem by establishing properly dispersive estimates.
Those results play a key role in the subsequent Section  where we prove the conservation of energy, momentum and angular momentum (Theorem REF ).
Section  is devoted to a numerical study on the comparison between classical wave equation and the present nonlocal problem and the outcome of this analysis essentially shows
that the peridynamic case seems to produce additional oscillations.
The paper ends with three Appendix sections in which some facts previously mentioned in the paper are specifically proved.
